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{"name": "Daniel's copy of Quantifiers 3-", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [{"variables": ["select"], "name": "Part 0"}, {"variables": ["select1"], "name": "Part 1"}, {"variables": ["select2"], "name": "Part 2"}], "parts": [{"layout": {"type": "all", "expression": ""}, "marks": 0, "matrix": "marks", "customMarkingAlgorithm": "", "type": "m_n_x", "showCellAnswerState": true, "minAnswers": "4", "maxAnswers": "4", "displayType": "radiogroup", "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "steps": [{"scripts": {}, "marks": 0, "type": "information", "useCustomName": false, "customMarkingAlgorithm": "", "showFeedbackIcon": true, "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "prompt": "<p>Consider the proposition:</p>\n<p><strong>There is someone whose name is not known to the rest of the group.</strong></p>\n<p>Using the predicate $P(m,n)$ we can express this as:</p>\n<p>$\\exists m \\forall n (\\neg P(n,m))$.</p>\n<p>To negate the proposition, we replace each $\\forall$ symbol with a $\\exists$ symbol, replace each $\\exists$ symbol with a $\\forall$ symbol, and write the negation of any statements:</p>\n<p>$\\forall m \\exists n (P(n,m))$.</p>\n<p>This corresponds to the English sentence (not asked for in this question):</p>\n<p><strong>Everyone's name is known by somebody else.</strong></p>", "customName": "", "variableReplacementStrategy": "originalfirst", "unitTests": [], "showCorrectAnswer": true}], "stepsPenalty": "0.25", "variableReplacementStrategy": "originalfirst", "minMarks": 0, "showCorrectAnswer": true, "scripts": {}, "unitTests": [], "useCustomName": false, "maxMarks": 0, "answers": ["{all[select[0]][1]}", "{all[select[1]][1]}", "{all[select[2]][1]}", "{all[select[3]][1]}"], "shuffleChoices": true, "showFeedbackIcon": true, "warningType": "warn", "choices": ["<span style=\"font-size: small;\">{all[select[0]][0]}</span>", "<span style=\"font-size: small;\">{all[select[1]][0]}</span>", "<span style=\"font-size: small;\">{all[select[2]][0]}</span>", "<span style=\"font-size: small;\">{all[select[3]][0]}</span>"], "prompt": "<p>If you want some help in answering this question click on Show steps. You will lose 0.25 marks as one of the questions is answered for you.</p>", "customName": "", "shuffleAnswers": false}, {"layout": {"type": "all", "expression": ""}, "marks": 0, "matrix": "marks", "customMarkingAlgorithm": "", "type": "m_n_x", "showCellAnswerState": true, "minAnswers": "4", "maxAnswers": "4", "displayType": "radiogroup", "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "minMarks": 0, "showCorrectAnswer": true, "scripts": {}, "unitTests": [], "useCustomName": false, "maxMarks": 0, "answers": ["{all[select1[0]][1]}", "{all[select1[1]][1]}", "{all[select1[2]][1]}", "{all[select1[3]][1]}"], "shuffleChoices": true, "showFeedbackIcon": true, "warningType": "warn", "choices": ["<span style=\"font-size: small;\">{all[select1[0]][0]}</span>", "<span style=\"font-size: small;\">{all[select1[1]][0]}</span>", "<span style=\"font-size: small;\">{all[select1[2]][0]}</span>", "<span style=\"font-size: small;\">{all[select1[3]][0]}</span>"], "customName": "", "shuffleAnswers": false}, {"layout": {"type": "all", "expression": ""}, "marks": 0, "matrix": "marks", "customMarkingAlgorithm": "", "type": "m_n_x", "showCellAnswerState": true, "minAnswers": "4", "maxAnswers": "4", "displayType": "radiogroup", "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "minMarks": 0, "showCorrectAnswer": true, "scripts": {}, "unitTests": [], "useCustomName": false, "maxMarks": 0, "answers": ["{all[select2[0]][1]}", "{all[select2[1]][1]}", "{all[select2[2]][1]}", "{all[select2[3]][1]}"], "shuffleChoices": true, "showFeedbackIcon": true, "warningType": "warn", "choices": ["<span style=\"font-size: small;\">{all[select2[0]][0]}</span>", "<span style=\"font-size: small;\">{all[select2[1]][0]}</span>", "<span style=\"font-size: small;\">{all[select2[2]][0]}</span>", "<span style=\"font-size: small;\">{all[select2[3]][0]}</span>"], "customName": "", "shuffleAnswers": false}], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>English sentences which are propositions are given&nbsp;and the appropriate logical expression chosen for the negation of the sentence.</p>"}, "functions": {}, "preamble": {"css": "", "js": ""}, "variablesTest": {"condition": "", "maxRuns": 100}, "name": "Daniel's copy of Quantifiers 3-", "rulesets": {}, "tags": [], "statement": "<p>In a seminar group, for group members $m$ and $n$, we let $P(m,n)$ &nbsp;be the predicate&nbsp;<strong><em>m knows the name of n&nbsp;</em></strong>.</p>\n<p><strong>Negate</strong> each of the following English sentences and choose the corresponding expression for the negated proposition involving quantifiers.</p>\n<p>Note that marks are deducted for each incorrect choice. However, the minimum mark is $0$.</p>", "variables": {"select2": {"description": "", "definition": "list(set(0..length(all)-1)-(set(select) or set(select1)))", "name": "select2", "templateType": "anything", "group": "Part 2"}, "select": {"description": "", "definition": "shuffle(list(0..length(all)-1))[0..4]", "name": "select", "templateType": "anything", "group": "Part 0"}, "marks": {"description": "", "definition": "list(0.5id(4)-0.25matrix(repeat(repeat(1,4),4)))", "name": "marks", "templateType": "anything", "group": "Ungrouped variables"}, "all": {"description": "", "definition": "[['There is someone whose name is not known to the rest of the group.',\n    '$\\\\forall m \\\\exists n (P(n,m))$',\n   'Everybody\\'s name is known by at least one other person.',\n   '$\\\\exists m \\\\forall n (\\\\neg P(n,m))$'],\n  ['Every group member doesn\\'t know the name of at least one other.',\n    '$\\\\exists m \\\\forall n ( P(m,n))$',\n   'Somebody knows the name of everybody.',\n  ' $\\\\forall m \\\\exists n (\\\\neg P(m,n))$'],\n  ['Nobody knows the name of anybody else.',\n    '$\\\\exists m \\\\exists n (P(m,n))$',\n   'At least one person knows the name of another.',\n   '$\\\\forall m \\\\forall n (\\\\neg P(m,n))$'],\n  ['There is a pair of group members who do not know each other\\'s name.',\n    '$\\\\forall m \\\\forall n (P(m,n) \\\\lor P(n,m))$',\n    'Given any pair of members, then at least one of them knows the name of the other.',\n    '$\\\\exists m \\\\exists n (\\\\neg P(m,n) \\\\land \\\\neg P(n,m)$'],\n  ['There is someone who knows everyone\\'s name.',\n    '$\\\\forall m \\\\exists n (\\\\neg P(m,n))$',\n    'Everybody doesn\\'t know the name of at least one other.',\n    '$\\\\exists m \\\\forall n (P(m,n))$'],\n  ['There is at least one person who knows the name of somebody else.',\n    '$\\\\forall m \\\\forall n ( \\\\neg P(n,m))$',\n    'Nobody knows the name of anyone else.',\n    '$\\\\exists m \\\\exists n (P(n,m))$'],\n  ['There is someone who doesn\\'t know the name of at least one other group member.',\n    '$\\\\forall m \\\\forall n (P(n,m))$',\n    'Everybody knows everybody else\\'s names.',\n    '$\\\\exists m \\\\exists n (\\\\neg P(n,m))$'],\n  ['Someone\\'s name is known to everyone else.',\n    '$\\\\forall m \\\\exists n ( \\\\neg P(n,m))$',\n    'Everybody\\'s name is not known by at least one other person.',\n    '$\\\\exists m \\\\forall n ( P(n,m))$'],\n  ['There is at least one person who does not know the name of anybody else.',\n    '$\\\\forall m \\\\exists n ( P(m,n))$',\n    'Everybody knows the name of at least one other.',\n    '$\\\\exists m \\\\forall n (\\\\neg P(m,n))$'],\n  ['Everybody knows at least one other person\\'s name.',\n    '$\\\\exists m \\\\forall n (\\\\neg P(m,n))$',\n    'There is somebody who does not know anyone else\\'s name.',\n    '$\\\\forall m \\\\exists n (P(m,n))$'],\n  ['Any member of the group has at least one person who doesn\\'t know their name.',\n    '$\\\\exists n \\\\forall m (P(m,n))$',\n    'There is someone whose name is known by everyone.',\n    '$\\\\forall n \\\\exists m (\\\\neg P(m,n))$'],\n  ['There are at least two people who know each other\\'s name.',\n    '$\\\\forall n \\\\forall m (\\\\neg P(n,m) \\\\lor \\\\neg P(m,n))$',\n    'For any two people in the group at least one doesn\\'t know the name of the other.',\n    '$\\\\exists n \\\\exists m (P(n,m) \\\\land P(m,n))$']\n  \n  ]", "name": "all", "templateType": "anything", "group": "Ungrouped variables"}, "select1": {"description": "", "definition": "list(set(0..length(all)-1)-set(select))[0..4]", "name": "select1", "templateType": "anything", "group": "Part 1"}}, "advice": "<p>In the following we use the rules for negating a proposition involving predicates and quantifiers as given in the lectures.</p>\n<p><strong>a)&nbsp;</strong></p>\n<p>1. The sentence:</p>\n<p style=\"text-align: center;\">&nbsp;<strong>{all[select[0]][0]}</strong>&nbsp;</p>\n<p>can be written in predicate form with quantifiers as:</p>\n<p style=\"text-align: center;\">&nbsp;&nbsp;<strong>{all[select[0]][3]}</strong></p>\n<p><strong></strong>The negation of the sentence can be written as:</p>\n<p style=\"text-align: center;\">&nbsp;<strong>{all[select[0]][2]}</strong>&nbsp;</p>\n<p>The predicate form with quantifiers for this is:</p>\n<p style=\"text-align: center;\">&nbsp;<strong>{all[select[0]][1]}</strong></p>\n<p style=\"text-align: left;\"><strong></strong></p>\n<p>2. The sentence:</p>\n<p style=\"text-align: center;\">&nbsp;<strong>{all[select[1]][0]}</strong>&nbsp;</p>\n<p>can be written in predicate form with quantifiers as:</p>\n<p style=\"text-align: center;\">&nbsp;&nbsp;<strong>{all[select[1]][3]}</strong></p>\n<p><strong></strong>The negation of the sentence can be written as:</p>\n<p style=\"text-align: center;\">&nbsp;<strong>{all[select[1]][2]}</strong>&nbsp;</p>\n<p>The predicate form with quantifiers for this is:</p>\n<p style=\"text-align: center;\">&nbsp;<strong>{all[select[1]][1]}</strong></p>\n<p style=\"text-align: left;\"></p>\n<p>3. The sentence:</p>\n<p style=\"text-align: center;\">&nbsp;<strong>{all[select[2]][0]}</strong>&nbsp;</p>\n<p>can be written in predicate form with quantifiers as:</p>\n<p style=\"text-align: center;\">&nbsp;&nbsp;<strong>{all[select[2]][3]}</strong></p>\n<p><strong></strong>The negation of the sentence can be written as:</p>\n<p style=\"text-align: center;\">&nbsp;<strong>{all[select[2]][2]}</strong>&nbsp;</p>\n<p>The predicate form with quantifiers for this is:</p>\n<p style=\"text-align: center;\">&nbsp;<strong>{all[select[2]][1]}</strong></p>\n<p style=\"text-align: left;\"><strong></strong></p>\n<p style=\"text-align: left;\">4. The sentence:</p>\n<p style=\"text-align: center;\">&nbsp;<strong>{all[select[3]][0]}</strong>&nbsp;</p>\n<p>can be written in predicate form with quantifiers as:</p>\n<p style=\"text-align: center;\">&nbsp;&nbsp;<strong>{all[select[3]][3]}</strong></p>\n<p><strong></strong>The negation of the sentence can be written as:</p>\n<p style=\"text-align: center;\">&nbsp;<strong>{all[select[3]][2]}</strong>&nbsp;</p>\n<p>The predicate form with quantifiers for this is:</p>\n<p style=\"text-align: center;\">&nbsp;<strong>{all[select[3]][1]}</strong></p>\n<p style=\"text-align: left;\">Similarly for Parts<strong> b)</strong> and <strong>c).</strong></p>\n<p style=\"text-align: left;\"><strong></strong></p>", "extensions": [], "ungrouped_variables": ["all", "marks"], "contributors": [{"name": "Daniel Mansfield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/743/"}, {"name": "Sean Gardiner", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2443/"}]}]}], "contributors": [{"name": "Daniel Mansfield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/743/"}, {"name": "Sean Gardiner", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2443/"}]}